Nonlinear Filters for Efficient Shock Computation

Engquist, Björn; Lötstedt, Per; Sjögreen, Björn

doi:10.2307/2008479

Cited by 32 publications

(27 citation statements)

References 14 publications

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“…corrections are made to each extremum in the characteristics Equations (5) and (9) can be written in the form arrays according to the criteria given in Algorithm 4.1 in [5]. The only modification we made is in the size of the correction ͳ (discussed in the next section).…”

Section: The Engquist Filtermentioning

confidence: 99%

“…Afterward, the physical vari-ѨU Ѩt ϩ ѨF Ѩx ϭ Q (13) ables are calculated from the corrected characteristic variables. As mentioned in [5], because TVD is enforced in characteristics the quasilinear form of which is space, the physical variables may occasionally violate the TVD criterion. Indeed, in our numerical experiments, TVD is occasionally violated in the physical variables; however, the oscilla-…”

Section: The Engquist Filtermentioning

confidence: 99%

“…Later in the decade, Engquist et al [5] proposed yet another method ofFor wave propagation problems in linear and bilinear elastic solids and a class of simple nonlinear solids the benefits of high-resolution introducing the required amount of dissipation, namely, the use schemes in gasdynamics can be immediately extended to solid of nonlinear filters. The advantage of these filters is that they mechanics.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Improved Solutions to the Small Strain Continuum Equations Using a Modified Engquist Filter

Lee¹,

Crawford²,

McDonough

1996

Journal of Computational Physics

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Section: The Engquist Filtermentioning

confidence: 99%

Section: The Engquist Filtermentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Improved Solutions to the Small Strain Continuum Equations Using a Modified Engquist Filter

Lee¹,

Crawford²,

McDonough

1996

Journal of Computational Physics

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“…In practice the use of a blend of low-and high-order dissipative terms with adaptive coefficients conditioned to the maximum local wave speed can yield effective shock capturing schemes. Recently Engquist, Lotstedt, and Sjogreen have shown that oscillations can be effectively controlled by post-processing the result after each time step with a filter which is applied only if a new extremum is detected (31). These procedures lead to a set of coupled ordinary differential equations, which can be written in the form dw dt…”

Section: Algorithms For Flow Simulationmentioning

confidence: 99%

Computational Aerodynamics for Aircraft Design

Jameson

1989

Science

105

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This article outlines some of the principal issues in the development of numerical methods for the prediction of flows over aircraft and their use in the design process. These include the choice of an appropriate mathematical model, the design of shock-capturing algorithms, the treatment of complex geometric configurations, and shape modifications to optimize the aerodynamic performance.While computational methods for simulating fluid flow have by now penetrated a broad variety of fields, including ship design, car design, studies of oil recovery, oceanography, meteorology, and astrophysics, they have assumed a dominant role in aeronautical science. In the aircraft industry there is often a very narrow margin between success and failure. In the past two decades the development of new commercial aircraft successful enough to make a profit for the manufacturer has proved an elusive goal. The economics of aircraft operation are such that even a small improvement in efficiency can translate into substantial savings in operational costs. Therefore, the operating efficiency of an airplane is a major consideration for potential buyers. This provides manufacturers with a compelling incentive to design more efficient aircraft.One route toward this goal is more precise aerodynamic design with the aid of computational simulation. In particular it is possible to attempt predictions in the transonic flow regime that is dominated by nonlinear effects, exemplified by the formation of shock waves. The importance of the transonic regime stems from the fact that to a first approximation, cruising efficiency is proportional to ML/D, where M is the Mach number (speed divided by the speed of sound), L is the lift, and D is the drag. As long as the speed is well below the speed of sound, the lift-to-drag ratio does not vary much with speed, so it pays to increase the speed until the effects of compressibility start to cause a radical change in the flow. This occurs when embedded pockets of supersonic flow appear, generally terminating in shock waves. A typical transonic flow pattern over a wing is illustrated in Fig. 1. As the Mach number is increased the shock waves become strong enough to cause a sharp increase in drag, and finally the pressure rise through the shock waves becomes so large that the boundary layer separates. The most efficient cruising speed is usually in the transonic regime just at the onset of drag rise, and the prediction of aerodynamic properties in steady transonic flow has therefore been a key challenge.Prior to 1965 computational methods were hardly used in aerodynamic analysis, although they were widely used for structural analysis. There was already in place a rather comprehensive mathematical formulation of fluid mechanics. This had been developed by elegant mathematical analysis, frequently guided by brilliant insights. Well-known examples include the airfoil theory of Kutta and Joukowski, Prandtl's wing and boundary layer theories, von Karman's analysis of the vortex street, and more recently Jo...

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“…For that reason, filtering methodŝ were developed, beginning in the late eighties. The first one, described by B. Engquist and B. Sjogreen in [1], uses simple TVD and conservation properties to correct nonphysical spurious oscillations from one time step to another. The correction step consists in pushing numerical data points up or down to an acceptable level while preserving conservation.…”

Section: Introductionmentioning

confidence: 99%

Essentially Nonoscillatory Postprocessing Filtering Methods

Lafon

Osher

1993

Algorithmic Trends in Computational Fluid Dynamics

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High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. In this paper, we present a new class of filtering methods denoted by ENO-LS (Essentially Nonoscillatory Least Squares) which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO framework.Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency and robustness of the method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters.

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Nonlinear Filters for Efficient Shock Computation

Cited by 32 publications

References 14 publications

Improved Solutions to the Small Strain Continuum Equations Using a Modified Engquist Filter

Improved Solutions to the Small Strain Continuum Equations Using a Modified Engquist Filter

Computational Aerodynamics for Aircraft Design

Essentially Nonoscillatory Postprocessing Filtering Methods

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